IB Math SL Functions and Transformations

Here you will find original exam style IB Math SL functions and transformations practice questions. Visit this page directly at hunkim.com/slfunctions

Key concepts

• Inverse functions
• Function composition
• Function transformations

1. Calculator allowed. f(x)=3x-2 and g(x)=x^4
   Find (f\circ g)(x)
   
2. See diagram below:
   
   1. Find f(5)
   2. Find f^{-1}(0)
   3. Domain of f^{-1}(x)?
   4. On the grid above, sketch the graph of f^{-1}
3. See graph below:
   
   1. Find f^{-1}(3)
   2. Find (f\circ f)(4)
   3. On the same diagram, sketch the graph of y=f(-x)
4. f(x)=(x+2)^3
   1. Find f^{-1}(x)
   2. Let g be a function so that (f\circ g)(x)=3x^6. Find g(x)
5. f(x)=6x+4 and g(x)=3x
   1. Write down g(3)
   2. Find (f\circ g)(x)
   3. Find f^{-1}(x)
6. Calculator allowed. f(x)=x^2+6x+9 and g(x)=x+5
   1. Find f(0)
   2. Find (g\circ f)(x)
   3. Solve (g\circ f)(x)=6
7. Calculator allowed. f(x)=x^2+6x+9 and g(x)=x+5
   1. Find f(0)
   2. Find (g\circ f)(x)
   3. Solve (g\circ f)(x)=6
8. f(x)=3x and g(x)=x^2+3
   1. Find f^{-1}(x)
   2. Find (f\circ g)(3)
9. Calculator allowed. f(x)=x^2-4 and g(x)=x^2-3
   1. Show that (f\circ g)(x)=x^4-6x^2+5
   2. On the following grid, sketch (f\circ g)(x)
      
   3. The equation (f\circ g)(x)=k has exactly two solutions. Find the possible values of k
      
10. The graph of y=x^2-3x is translated 3 units to the right. Find the resulting equation in the form y=ax^2+bx+c
    
11. Describe a transformation which transforms the graph of y=\log x+1 to the graph \log x^3+3
    
12. Point (-3,2) is on f(x). g(x)=f\left(\frac{2x-1}{3}\right). What point must be on g(x)?
    
13. Find two transformations whose composition transforms the graph y=(x-2)^2 to the graph of y=2(x+1)^2
    
14. f(x)=2x^2-2x+1. The graph of g is obtained by reflecting f on the y-axis, followed by a translation of \begin{pmatrix}2 \\ 3\end{pmatrix}. What are the coordinates of the vertex of g(x)?
    
15. See f(x)=\log_k(x-2) below:
    
    1. Find k
    2. g(x)=f^{-1}(x)-1. What is the range of g(x)?
       
16. See f(x) in the diagram below:
    
    1. Evaluate f^{-1}(0)
    2. Sketch on the same graph y=f^{-1}(-x)
       
17. f(x)=e^{\ln x^2+\ln 4}. g(x)=f\left(\frac{x}{2}-2\right). Sketch g(x)
    
18. h(x)=\frac{3x+2}{x-5}
    1. What is the vertical stretch factor of h(x) as compared to f(x)=\frac{1}{x}?
    2. What are the coordinates of where the asymptotes of h^{-1}(x) meet?
    3. If we shift h^{-1}(x) with a translation vector of \begin{pmatrix}k \\ 0\end{pmatrix}, the new function becomes a self-inverse function. Find k